Shapes and Sampling

MicrostateShape

Abstract supertype defining the basic structure and layout of a microstate (or motif).

A MicrostateShape specifies which relative positions are retrieved from the data to evaluate recurrences, and how these binary recurrence values are interpreted and mapped to a decimal representation for counting.

All subtypes of MicrostateShape must include a field expr, which defines the RecurrenceExpression used to compute recurrences.

Implementations

• Diagonal
• Rect
• Triangle

Rect <: MicrostateShape

Define a rectangular microstate shape.

Rect can represent either a two-dimensional microstate (identified as Rect2, used for Recurrence Plots and Cross-Recurrence Plots) or an N-dimensional microstate (identified as RectN, used for spatial data).

Rect2 (time-series data)

A 2D rectangular microstate can be initialized using either of the following constructors:

Rect(expr::E; rows = 2, cols = 2, B = 2)
Rect(rows::Int, cols::Int; expr = Standard(0.27), B = 2)

Here, rows and columns define the rectangle dimensions, expr is the RecurrenceExpression used to evaluate recurrences, and B is the base used to encode microstate elements (typically 2, representing recurrence or non-recurrence).

Rectangular microstates can be specialized to define common patterns such as lines, columns, and squares:

line = Rect(expr; rows = n, cols = 1)
column = Rect(expr; rows = 1, cols = n)
square = Rect(expr; rows = n, cols = n)

Since square microstates are frequently used, a convenience constructor is also provided:

Rect(expr::E, N; B = 2)

where N defines the size of the square microstate. For example:

square = Rect(expr, n)

RectN (spatial data)

For N-dimensional structures, typically used with spatial data, the RectN variant can be initialized as:

Rect(expr::E, structure::NTuple{D, Int}; B = 2)

Here, structure defines the size of the microstate along each dimension. For example:

nrect = Rect(expr, (2, 1, 2, 1))

This form is suitable for N-dimensional spatial data, such as images or volumetric datasets.

Triangle{N, B, E<:RecurrenceExpression} <: MicrostateShape

Triangle{N, B, E<:RecurrenceExpression} <: MicrostateShape

Define a triangular microstate shape, originally introduced by Hirata in 2021 (Hirata, 2021).

Constructor

Triangle(expr::E, N::Int; B::Int = 2)

where expr is the RecurrenceExpression used to evaluate recurrences and N defines the size of the triangular microstate.

Example

n = 3
triangle = Triangle(expr, n)

The corresponding microstate structure is given by:

\[\begin{pmatrix} R_{i,j} & R_{i,j + 1} & R_{i,j + 2} \\ & R_{i + 1,j + 1} & R_{i + 1,j + 2} \\ & & R_{i + 2,j + 2} \\ \end{pmatrix}\]

Diagonal <: MicrostateShape

Define a diagonal microstate shape, which captures recurrences along diagonals of a Recurrence Plot (RP).

Constructor

Diagonal(expr::E, N::Int; B::Int = 2)

where expr is the RecurrenceExpression used to evaluate recurrences and N defines the length of the diagonal microstate.

Example

diagonal = Diagonal(expr, 3)

SamplingMode

Abstract supertype defining how the initial position $(i, j)$ of each microstate is selected during the construction of recurrence microstate distributions.

Implementations

• SRandom
• Full

SamplingSpace

Define the range of valid indices used to sample the initial positions $(i, j)$ of microstates.

SRandom{F<:Real} <: SamplingMode

Sampling mode that randomly selects microstate positions $(i, j)$ within the SamplingSpace.

Constructors

SRandom(num_samples::Int)
SRandom(ratio::Union{Float32, Float64})

The sampling mode can be initialized either by specifying the exact number of microstates to sample or by providing a ratio of the total number of possible microstates.

Examples

s = SRandom(1000)   # Specify the exact number of sampled microstates
s = SRandom(0.05)   # Specify a ratio of the total possible microstates

Full <: SamplingMode

Sampling mode that selects all possible microstates within the SamplingSpace.

Constructor

s = Full()